A general treatment for the deduction theorem in open calculi

Arthur Buchsbaum, Tarcisio Pequeno

Abstract

Aiming clear formulations of the deduction theorem and precise account of its restrictions, a study concerning the rules introducing connectives related to its behavior is conducted. Special attention is devoted to operators, such as quantifiers and modalities, dealing with varying objects. Concepts and techniques able to cope with the subleties of tracing those objects are introduced. As a result, a classification of calculi in terms of robustness to applications of the deduction theorem is given.